The author of these
remembrances (from now on only the 'author') feels painfully that he is in an awkward
position. He intends to show a side of Warren McCulloch which is not very well - if it all
- known and which hardly becomes visible in the publications of this very great man and
first rate scientist: we refer to his importance and profundity as a philosopher. He was
aware and very intensely so - of Cybernetics as a discipline sui generis that needed a
novel philosophic foundation to distinguish if from the conventional disciplines. This
conviction of his finally led to the meeting with the author - a contact which lasted
almost a decennium. The quandary the author finds himself in stems from the fact that he
entertained and still entertains almost identical views about the relation between
cybernetics and philosophy as McCulloch and finds it therefore almost impossible to
perform a clean separation of his own ideas from those of McCulloch. He is only sure that
the thoughts he expressed on cybernetic topics are fully his own up to the publication of
his "Cybernetic Ontology and Transjunctional Operations" which came out in 1962.
Although McCulloch is already quoted in this essay it was done solely with the intent to
appeal to his authority for ideas which the author had entertained for quite a while.

The contact between the author and Warren McCulloch was
established after Dr. John Ford, then at the George Washington University, had given
McCulloch in 1959 a German paper of the author "Die aristotelische Logik des Seins
und die nicht-aristotelische Logik der Reflexion" which had come out in Germany in
1958. He is still intensely grateful to Dr. Ford for having made this connection which was
bound to change his total outlook on philosophy. However, it took some time before he
really understood what had attracted Warren McCulloch to his paper. It was not so much its
potential applicability to cybernetics but a hidden relation that it revealed between
number and logical context. When the author wrote it he opined that a non-Aristotelian
Logic is nothing but a place value system of innumerable logical sub-systems of
Aristotelian (two-valued) character. His interest was at that time wholly conceptual and
he did not even dream that a hidden arithmetical issue might lead into deeper foundational
layers of Cybernetics. Here McCulloch was far ahead of him.

Their intellectual collaboration started
in earnest when some evening the author had made a stop-over on his yearly trip to New
Hamsphire - McCulloch led the talk to the Pythagoreans and their theorem that numbers
describe the ultimate core of Reality. Although the author pressed for a detailed
explanation all he was told at that time was that to find out more was exactly his own
business. It was the first time that the author encountered a peculiar reticence of
McCulloch's regarding ontological or - more precisely - 'metaphysical' questions. It led
him to grossly underestimate McCulloch's gifts and intuitions in this direction. He was
confirmed in his faulty judgment when he noticed that McCulloch never bothered to make
corrective remarks when a paper which was read at a congress or sympo- sion where he was
present obviously implied metaphysical assumptions which had to be partly or totally
wrong. First he assumed that McCulloch was not aware of it; later however the author knew
better. Nevertheless he must confess that during the whole duration of his acquaintance
and - as the author hopes friendship McCulloch never gave up his reluctance to criticise
the course cybernetics was taking with relation to Philosophy. Only after McCulloch's
death he learned that his mentor in Cybernetics had been as dissatisfied as he himself
with the lack of fundamental ontological orientation that characterized - and still
characterizes - the pursuit of cybernetic theories. But he came to understand very soon
how much McCulloch saw his own endeavours within a novel metaphysical frame. The
revelation came one evening when McCulloch started to talk about Martin Heidegger and
produced a copy, very shabby and dilapidated from intensive use, of "Sein und
Zeit".

The book had originally belonged to his
friend and coworker Eilhard von Domarus, so he explained; he in his turn had studied it
carefully and he now wanted to give it to the author for renewed study because the latter
had confessed that he did not care very much for Heidegger's philosophy. The expression of
thanks for the unexpected present must have sounded rather reluctant because McCulloch
grew very eloquent and insisted that the "Nichts" (Nought) in Heidegger's
philosophy was precisely the ontological locus where the central problem of cybernetics
was located, namely the mapping of the process of Life onto matter per se inanimate. BEING
is both: subject and object as well; but western philosophy has fallen into
"Seinsvergessenheit" (oblivion of ultimate Reality) since the time of the Greek.
Which in McCulloch's view meant: it did not focus on the problem of cybernetics. In
classic philosophy mere objectivity without self-reference is mistaken for
"Sein". When McCulloch commented on Heidegger with these remarks the author knew
he had underestimated his philosophical gifts. His detailed knowledge of "Sein und
Zeit" and especially his discussion of this "Nichts" gave the author's
metaphysical thinking a new direction and made him look for the roots of Cybernetics in
the ultimate and primordial recesses of the Universe.

Since the spiritual contact point between
MeCulloch and the author happened to be their common interest in the transcendental
relevance of logic in other words: how much and what information logic conveys about the
world that surrounds us - it was only natural that the author wanted to know from his
partner what he meant by the term 'metaphysical'. For a start he was referred to the
"Mysterium Iniquitatis ..." and the notions that "prescribe ways of
thinking physically about affairs called mental ..." It stands to reason that this
answer left the philosopher dissatisfied and it surely did not cover McCulloch's own -
very ambivalent appreciation - of Heidegger. This was admitted; and then MeCulloch started
to express thoughts which went far beyond the metaphysical references imbedded in papers
like the "Mysterium Iniquitatis" "Through the Den of the
Metaphysician", "What is a Number ..." and others. He drew the author's
attention to the fact that any logic or calculus Man may ever conceive is nothing but a
more or less competent formalization of ontological concepts. This ideas was, of course,
not new and may be easily extracted from his writings as ever present implication. But it
showed that he had wandered much deeper into the grottoes of metaphysics than he was
inclined to express explicitly in his papers. At this juncture the author thinks it
fitting to remind the reader of the quotation of Clerk Maxwell appearing in "Through
the Den of the Metaphysician" about the relation between thoughts and the molecular
motions of the brain: "does not the way to it lie through the very den of the
metaphysician, strewn with the bones of former explorers and abhorred by every man of
science?" McCulloch comments this quotation with a "Let us peacefully answer the
first half of this question 'Yes', the second half No', and then proceed serenely."

While there can be no doubt that he never
abhorred the den of metaphysics his texts show a pronounced reluctance to analyze in
detail the accoutrements of Transcendence. On the other hand, this reluctance disappeared
almost completely when speculating on the pertinent issues in the presence of a person who
was much more at home in the realms of the Transcendental than in the empirical ways of
Cybernetics as happened to be the case with the author.

From Heidegger's "Nichts" the
discourse went to Kant and Hegel. The author must confess that he was somewhat surprised
when he discovered that McCulloch understood that Kant's philosophy closes an epoch of
philosophical thought and that Hegel opens a new one. He knew this, of course, himself, -
that was after all his business - but he had interpreted it in terms of the distinction
between 'Natur- and Geisteswissenschaft' and the pseudo-systematic development of the
latter in the Hegel-Renaissance since 1900. Of the Hegel-Renaissance and its concomitant
intellectual events McCulloch was hardly aware. Even if he had been familiar with it: the
metaphysical gap between matter and mind or subject and object which was emphasized by the
Geisteswissenschaft could not be accepted by any cyberneticist, least of all McCulloch.
Consequently, he explained the distinction between Kant and Hegel by pointing out the
different view of Dialectics entertained in the Critique of Pure Reason and in Hegel's
Logic. Kant deals with Dialectics in the sense of the Platonic tradition and in the
Critique of Pure Reason the dialectic argument ends in the transcendental illusion as the
unavoidable admixture of error that infiltrates all metaphysical assertions. Thus Kant's
evaluation of Dialectics is basically negative and the less we imbibe of this poisonous
drink the better off we are. For Hegel, on the other hand, he explained, the dialectic
structure is a legitimate element of thought as well as of objective existence and it
furnished the transcendental link that connects both. Seymour Papert has referred to this
situation when he reports in his Introduction to the Embodiments of Mind that McCulloch
insisted "that to understand such complex things as numbers we must know how to
embody them in nets of simple neurons. But he would add that we cannot pretend to
understand these nets of simple neurons until we know - which we do not except for an
existence proof - how they embody such complex things as numbers. We must, so to speak,
maintain a dialectical balance between evading the problem of knowledge by declaring that
it is 'nothing but' an affair of simple neurons, without postulating 'anything but'
neurons in the brain. The point is, if I understand him well, that the 'something but' we
need is not of the brain but of our minds.. namely, a mathematical theory of complex
relations powerful enough to bridge the gap between the level of neurons and the level of
knowledge in a far more detailed way than can any we now possess." (p. XIX)

After the author had read this
introduction he asked McCulloch whether he really intended to introduce dialectics only in
a loose and logically non-coercive manner or whether he realized that Hegel employed the
term as a linguistic cover for a hidden exact mechanism which the Universe as a whole
employed but which we were still incapable of unravelling. McCulloch remained silent for a
few moments and then asked the author to rephrase the question, which the latter did by
simply inquiring whether he thought that the term 'dialectics' merely referred to a quirk
or weakness of the human mind or whether it indicated an intrinsic property of Reality.
This time McCulloch answered that the term should designate an objective quality of the
universe and he added: I think this is what separates Kant from Hegel. The author and
McCulloch agreed that the "so to speak" in the lengthy quotation above was not a
proper expression because it suggested only a vague analogy. It did not indicate that in
the term "dialectical" a very precise systematic foundation problem of
mathematical theory was at hand.

The author cannot now remember how the
talk got to a paper of Barkley Rosser "On Many-Valued Logic", which was
published in the American Journal of Physics (Vol.9,4; pp. 207-212, 1941), and from there
to the question whether a dialectical analysis of natural numbers might help to bridge the
gap between the level of neurons and the level of knowledge which is conveyed by present
mathematical theory. Everything was still very vague, and it took an almost nightlong
discussion to clear the realm of discourse somewhat. It helped greatly that McCulloch was
familiar with the distinction of number by Plato and Aristotle and how much nearer to the
Pythagoreans Plato's ideas were than those of Aristotle. And then he surprised the author
by saying that, what Hegel meant by number was a not very successful attempt to rebuild
again the general concept of numerality which had been divided by the antagonism of
Platonic and Aristotelian philosophy. He finally added that Hegel failed to develop a
novel theory of mathematical foundation because he thought more about number in the
Aristotelian than in the Platonic sense. This was a most astounding conclusion and seemed
questionable to the author. He believed that he knew more about Hegel and felt unable to
accept McCulloch's thesis. Since the whole history of mathematics from the Greeks to the
present time owes all its success to the instinctive acceptance of the Aristotelian way of
thinking ahout numbers McCulloch had to be wrong. The author left Shady Hill Square
somewhat dissatisfied and went skiing.

Six weeks later he was back, very
contrite and humble. He was not a mathematician, only a logician, moreover reared in the
atmosphere of the Geisteswissenschaften. But it had, in the meantime, dawned upon him how
much hetter a philosopher McCulloch was when the mind turned to the problem of the
transcendental relation between mathematics and the Universe. Conceding McCulloch his
Hegel interpretation the discussion doubled back to the essay of Barkley Rosser. Rosser's
attempt seemed now extremely interesting; Rosser had demonstrated in his paper, that one
can get numbers from four ideas in two-valued logic which have been formalized in terms of
a likewise two-valued calculus. The first idea is 'conjunction' (... and ...); the second
idea is 'negation' (not ...); the third idea is 'all'; and the final idea is 'is a member
of'. Rosser then suggests a projection of these ideas onto the structure of a many-valued
calculus. For the purpose of demonstration and to retain a comparative simplicity he
exemplifies his case with a three-valued logic. As values he chooses 'true' (T),
'probable' (?), and 'false' (F). McCulloch and the author agreed that this interpretation
of three-valuedness has proved its usefulness in cybernetics and elsewhere but that it
could not lead to a trans-classic theory of natural numbers because it has been
established since at least 1950 (Oskar Becker) that the introduction of probability or
modal values destroys the formal character of a logical system. For if strict formality is
insisted on any such spurious many-valued system reduces itself automatically to a
two-valued calculus. In order to convince McCulloch that Rosser's approach to the problem
needed a weighty correction the author pointed to something which he considered Rosser's
second mistake. The latter determines conjunction in classic logic by the following
matrix:

T

F

T

T

F

F

F

F

and the stipulation that T
is not permitted to re-occur in any of the empty places which originate if we extend the
places for the functional result from 4 to 9. Thus he defines, in strict analogy,
three-valued conjunction by the matrix:

T

?

F

T

T

.

.

?

.

.

.

F

.

.

.

We repeat: in order to
retain the meaning of conjunction T is not to go in any of the empty places which are left
open in the above matrix. However (?) and (F) may go indiscriminately in any of the other
squares. Since 8 squares are left to be filled and since two choices are available in the
case of each square there are 28, i. e. 256 possible choices for filling the squares. in
Rosser's opinion all of them represent the general meaning of conjunction in a
three-valued logic. This claim was easily refutable if one recognized - as McCulloch did -
the interpretation of trans-classic logic as given by the author in his "Cybernetic
Ontology and Transjunctional Operations". In order to demonstrate Rosser's too
generous interpretation of conjunction the author filled out the matrix in the following
way:

1

2

3

1

1

3

3

2

3

2

3

3

3

3

2

In order to avoid the
ontological consequences which are implied in Rosser's use of the symbols T for truth, ?
for probability or modality, F for false we have denoted the values in the same order with
the first three integers. This choice of values is quite in accordance with Rosser's
stipulation for the meaning of conjunction. However, there it not the remotest chance to
interpret this arrangement as a matrix of a conjunctive functor. To render a minimum sense
of conjunction a three-valued logic would have to retain the structural feature of
conjunctivity in at least one of the two-valued alternatives 1 or 2,2 or 3, or 1 or 3.
This is not be case, because or the two-valued system encompassing the first and the
second value we obtain the morphogrammatic structure which can only be filled by
trans-junctional value-occupancy. For the two-valued system constituted by 2 and 3 we
obtain a morphogrammatic structure for value-occupancy which is demanded in the case of
equivalence, and for the final two-valued system the morphogrammatic structure of
transjunction re-occurs.

But let us, for argument's sake, assume
that Rosser is right and we have to deal with 256 possible kinds of conjunction in a
three-valued system. What shall we do with this embarrassing wealth? Rosser himself gives
the answer: "Apparently the only thing that can be done about the matter is to pick
out the 'and' that one likes best, and try to ignore the rest. " Emphasis by
G. G.). McCulloch pointed out that the arbitrariness which Rosser suggested could not be
tolerated in the development of a more basic theory of natural numbers. But he added
meditatively: It hints at something in the relation between matter and form. The author is
not quite clear whether this was McCulloch's exact wording; at any rate, he asked his
mentor what he meant and McCulloch spun a long tale which seemed to the hearer to go far
beyond what he had learned from the essay' "What is a Number that Man may know it
...?". Finally a spark of tentative understanding jumped from the speaker to the
listener. McCulloch was talking about Hermeneutics and about the possibility that, if
numbers were subject to hermeneutic procedures in the sense of Dilthey's 'Verstehen' in
the Geisteswissenschaften, this would definitely close for the scientist the gap between
Nature and Geist. The idea of a basic 'arithmetization' of the Geisteswissenschaften
seemed to the author at that time not only bizarre but outrageous and he voiced his
violent objections. McCulloch did not answer any of them; he only asked curtly: and what
do you make of Rosser's "sidewise motion"? (The reader who is not familiar with
this paper should be informed that Rosser said in his somewhat loose manner that the
mapping of natural numbers on a many-valued logic produces something like a "sidewise
motion" of these numbers.)